\documentclass[t,12pt,aspectratio=169]{beamer} % 16:9 宽屏比例，适合现代投影
\usepackage{ctex} % 中文支持
\usepackage{amsmath, amssymb, amsfonts} % 数学公式与符号
\usepackage{graphicx}
\usepackage{multirow, multicol}
%\usepackage{hyperref, url}
%\usepackage{enumitem}
\usepackage{booktabs}
\usepackage[font=small,labelfont=bf]{caption}

% 主题设置（推荐简洁风格）
\usetheme{Madrid}
\usecolortheme{default} % 可选：seahorse, beaver, dolphin 等

\author{王立庆（2024级数学与应用数学1班）}
\title{常微分方程教学大纲}
%\date{2025年9月1日}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 目录页
\begin{frame}{目录}
 \tableofcontents
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{时间地点}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{时间地点}

\begin{enumerate}\itemsep1em 
\item 上课时间地点：周二下午5-6节，六教102；周四上午3-4节，六教102.
\item 答疑时间地点：周二下午7-8节, 周二晚上19:00 - 20:30, 一教210. 
\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{参考教材}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{参考教材}

\begin{enumerate}\itemsep1em 
\item 丁同仁、李承治，常微分方程教程，高等教育出版社，2022年3月第三版。

%\item  柳彬，常微分方程，北京大学出版社，2023年10月第一版。
\item  王高雄、周之铭、朱思铭、王寿松，常微分方程，高等教育出版社，2020年7月第四版。

\item  C. Henry Edwards, David E. Penney, 常微分方程基础(第5版)，机械工业出版社，2020年2月。

\item  Vladimir I. Arnold. 阿诺德常微分方程讲义. 世界图书出版公司. 2024年3月。

\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{教材目录}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{教材目录}

\begin{itemize}\itemsep0.2em 
\item  一、基本概念 
\item  二、初等积分法 
\item  三、存在唯一性 
\item  四、奇解 
\item  五、高阶微分方程 
\item  六、线性微分方程组 
\item  七、幂级数解法 
\item  八、定性理论分支理论 
\item  九、边值问题 
\item  十、首次积分 
\item  十一、一阶偏微分方程 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{课程成绩}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{课程成绩}
\begin{enumerate}\itemsep0.8em 
\item  平时成绩 50\%.
\begin{enumerate}\itemsep0.5em 
\item[1.1.] 课堂测验10次，共20分。课堂书面完成。

\item[1.2.] 课外作业10次，共20分。学习通作业完成。

\item[1.3.] 课程论文1篇，共20分。选择一个常微分方程的重要定理。阅读一篇相关文献。撰写一篇课程小论文和一个学术报告PPT。学习通作业完成。

\item[1.4.] 案例分析1篇，共20分。选择一个常微分方程模型，使用观测或模拟数据拟合微分方程模型的参数。学习通作业完成。%了解数学建模竞赛中的使用微分方程模型的一些题目。

\item[1.5.] 期中考试1次，共20分。在教室里书面闭卷考试。
\end{enumerate}

\item  期末成绩 50\%, 计算题和应用题。

\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{常微分方程的基本概念}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{常微分方程的概念}

\begin{multicols}{3}
\begin{enumerate}\itemsep0em 

\item 微分方程  
\item 常微分方程  
\item 阶数  
\item 解（通解、特解）  
\item 初值问题  
\item 积分曲线  
\item 方向场(线素场)  
\item 存在唯一性  
\item Lipschitz条件  
\item 分离变量法  
\item 积分因子  
\item 恰当微分方程  
\item 一阶线性方程  
\item 齐次方程%（可化为变量分离）  
\item 贝努利方程  
\item 奇解  
\item 包络  
\item Clairaut方程  
\item 高阶微分方程  
\item 线性相关与线性无关  
\item Wronsky行列式  
\item 齐次线性方程  
\item 非齐次线性方程  
\item 特征方程  
\item 常数变易法  
\item 待定系数法  
%\item 微分方程组  
\item 基本解组  
\item 相平面与相图  
\item 平衡点与稳定性	
\item 幂级数解法  
\item 正则奇点%与Frobenius方法  
%\item 矩阵指数  
%\item 非线性系统的平衡点  
\item 极限环  
\item Liapunov函数  
%\item 分支（Bifurcation）现象  
\item 边值问题  
\item Green函数  
\item Sturm-Liouville问题  

\end{enumerate}
\end{multicols}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{常微分方程的求解方法}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{常微分方程的求解方法}

\begin{multicols}{2}
\begin{enumerate}\itemsep0em 

\item 分离变量法  
\item 积分因子法  
\item 变量代换法  
\item 微分法%（恰当方程求解）  
\item 参数法  
\item 幂级数方法  
\item 常数变易法（常数变异法）  
\item 皮卡序列（Picard迭代法）  
\item 欧拉方法（Euler方法）  
\item 特征方程法  
\item 待定系数法  
\item 降阶法（如不显含 $y$ 或 $x$）% 的高阶方程）  
\item 算子法（微分算子法）  
\item Laplace变换法  
\item 矩阵指数法（用于线性系统）  
\item 特征线法（用于一阶偏微分方程）  
\item Frobenius方法（正则奇点附近的级数解）  
\item 数值积分法（如Runge-Kutta方法）  
\item 平衡点分析与线性化方法（定性求解）  
\item 首次积分法（利用守恒量降阶）
\end{enumerate}
\end{multicols}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{常微分方程的应用例子}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{常微分方程的应用例子}

\begin{multicols}{3}
\begin{enumerate}\itemsep0em 

\item  自由落体运动  
\item  条形磁铁的磁场  
\item  特技跳伞  
\item  RL串联电路  
\item  人口数量（Malthus）  
\item  捕食者-被捕食者模型（Lotka-Volterra）  
\item  追线问题%（狗追人）  
\item  逃逸速度  
\item  传染病模型（SIR）  
\item  单摆运动  
\item  悬链线  
\item  二体问题（行星轨道）  
\item  弹簧振子（无阻尼）  
\item  牛顿冷却定律  
\item  放射性衰变  
\item  混合问题（盐水罐）  
\item  RLC串联电路  
\item  Logistic人口模型  
\item  一级化学反应  
\item  梁的弯曲（Euler-Bernoulli）  
\item  最速降线问题  
\item  等角螺线（对数螺线）  
\item  火箭推进（变质量）  
\item  神经元模型（LIF）  
\item  药物动力学（单室模型）  
\item  行星轨道的径向运动（有效势能）  
\item  范德波尔振子  
\item  一维稳态热传导

\end{enumerate}
\end{multicols}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{常微分方程的重要定理}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{常微分方程的重要定理}
\begin{enumerate}\itemsep0.3em 
\item Cauchy-Lipschitz定理（解的存在唯一性定理）  
\item Peano存在性定理  
\item 解的延拓定理  
\item 线性微分方程组解的叠加原理  
\item Wronsky行列式定理  
\item 常系数齐次线性微分方程的特征方程定理  
\item 非齐次线性微分方程通解结构定理  
\item Liapunov稳定性定理（第二方法）  
\item Sturm比较定理  
\item Poincaré-Bendixson定理
\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setlength{\parskip}{1em} % 后面每页设置段间距为 1em
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{每周授课计划}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第1周}

%1
1. 基本概念： 微分方程的定义、阶数、通解、特解；初值问题；方向场、线素场、积分曲线。

2. 实际例子：落体运动、条形磁铁的磁力线。

3. 求解方法：分离变量法。

4. 重要定理：无。

5. 作业：抄写两个实际例子。1.1: 1(1), 2(1)。 1.2: 1(1), 2(1)。

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第2周}

%3 一阶方程：齐次与恰当方程 齐次方程变换；恰当微分方程与积分因子。
(2.2) 变量分离方程： 例1、例2、例3.

(2.3) 一阶线性方程： 例1、例2、例3. 

实际例子：RL电路、追线问题。

作业：抄写两个实际例子。2.2: 1(1), 2(1), 3(1)。 2.3: 1(1), 2(1)。 

%一阶线性方程：一阶线性方程的积分因子法；常数变易法；应用（RL电路、药物动力学）。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第3周}

(2.4) 初等变换法： 例1、例2、例3、伯努利方程、里卡蒂方程。

(2.6) 应用举例： 例1（正交轨线族）、例2（人口模型）、例3（捕食者被捕食者）. 

求解方法：变量代换。

实际例子：探照灯镜面、人口问题。

作业：抄写两个实际例子。2.4: 1(1), 2(1), 3(1)。 2.6: 1(1), 2(1)。 

%3  
% 5 一阶方程综合与建模： 复杂一阶方程求解技巧；追线问题、特技跳伞模型。

% 6 解的存在唯一性（局部）： Cauchy-Lipschitz定理；Lipschitz条件；Peano存在性定理（简述）。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第4周}
%4 
% 7 解的延拓与连续依赖性： 解的延拓定理；解对初值的连续依赖性。

% 8 一阶隐式方程： 参数法求解 $ F(y', y, x) = 0 $；Clairaut方程。

复习：第1章：第1、2节。第2章：第2、3、4、6节。

作业习题：实际例子2.6.3. （捕食者被捕食者）、习题2.6.5.（逃逸速度）、2.6.6.（传染病）。

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第5周}
  
%5 
9 二阶线性方程 - 齐次情形： 线性无关性、Wronsky行列式；通解结构；常系数齐次方程（特征方程法）。

10 二阶线性方程 - 非齐次情形： 非齐次通解结构；常数变易法；待定系数法（多项式、指数、三角函数）。

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第6周}

%6 
11 高阶线性方程： 高阶线性方程的通解结构；常系数情形；应用（弹簧振子、RLC电路）。

12 振动模型分析： 无阻尼与阻尼振动；强迫振动与共振；机电类比（质量-弹簧-阻尼 $\leftrightarrow$ RLC电路）。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第7周}

%7 
13 幂级数解法： 在常点邻域的幂级数解；Legendre方程（简例）。

14 期中复习： 知识点梳理；典型题型讲解；答疑。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第8周}

%8 
15 期中考试： 考试（覆盖第1-7周内容）

16 考试讲评与反馈： 试卷分析。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第9周}
%9 
17 线性微分方程组 - 基本理论： 一阶线性系统；解的存在唯一性；齐次系统解空间结构；Wronsky行列式。

18 常系数线性系统 - 对角化法： 特征值与特征向量法；实特征值情形；相图分析（节点、鞍点）。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第10周}

%10 
19 常系数线性系统 - 复特征值： 复特征值对应的螺旋点、中心；相图绘制。

20 矩阵指数法 - 矩阵指数： $ e^{At} $ 的定义与计算；非齐次系统的常数变易法。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第11周}

%11 
21 非线性系统 - 平衡点与线性化： 平衡点分类；Jacobi矩阵；线性近似；稳定性判据。

22 非线性系统 - 相平面分析： Lotka-Volterra模型相图；极限环初步；范德波尔振子。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第12周}

%12 
23 首次积分与守恒律：首次积分定义；机械能守恒（单摆、二体问题）；角动量守恒。

24 几何应用模型：悬链线；最速降线；等角螺线。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第13周}

%13 
25 边值问题初步： 二阶边值问题；特征值与特征函数；简单例子（热传导、梁弯曲）。

26 稳定性理论（Liapunov方法）： Liapunov函数定义；渐近稳定性判别法；应用举例。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第14周}

%14 
27 综合建模专题： SIR传染病模型；火箭推进；神经元模型。

28 数值方法简介： Euler法、改进Euler法、Runge-Kutta法思想与实现（结合Python演示）。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{学习计划第15周}

%15 
29 课程总复习： 知识体系梳理；重点难点回顾；典型综合题讲解。

30 课程总结与展望： 总结微分方程的思想方法；介绍偏微分方程、动力系统等后续方向。


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

